## Showing $supp{f} =\overline{\{x \in X: f(x) \neq 0\}}$

Let $$f: \mathbb R^{d} \to \mathbb R$$

We define the support of $$f$$ as $$supp f=(\bigcup_{A \subseteq \mathbb R^{d} open,f\vert_{A}=0, \lambda^{d}-a.e.}A)^{c}$$

How can I show for $$f \in C(\mathbb R^{d})$$ that $$supp{f} =\overline{\{x \in X: f(x) \neq 0\}}$$

How do I go about showing $$(\bigcup_{A \subseteq \mathbb R^{d} open,f\vert_{A}=0, \lambda^{d}-a.e.}A)^{c}\subseteq \overline{\{x \in X: f(x) \neq 0\}}$$

and conversely,

$$\overline{\{x \in X: f(x) \neq 0\}}\subseteq (\bigcup_{A \subseteq \mathbb R^{d} open,f\vert_{A}=0, \lambda^{d}-a.e.}A)^{c}$$